One could say that in a way, the religious missions were a sort of 'invasion' of Japan.

Not all challenges to power are done through purely military means.

As we all know today, economic, political, cultural, and indeed, religious/spiritual conflict can sometimes be as much of a threat as any army or navy. And during the contemporary times of when European missionaries were making large efforts to save Japanese souls for God, religion, politics, economics, and culture were intricately intertwined.

But before we go any further we should directly address your question and say, no, there were no clear or organized plans for Christian missionaries to pave the way for a military invasion, nor were they secretly trying to overthrow the Shogunate as an end game plan.

The question is then, why did the missions receive so much backing from not only religious leaders/organizations but also royalty, nobility, guilds, merchants, etc?

One of the big reasons is also the most in your face.

Spreading Christianity was a good deed, as was bringing salvation to the ignorant, especially if it increased the influence and sphere of influence of your sect/church/religion

The various Christian orders that sent missionaries to Japan were obviously very devoted to the cause of spreading the faith and many of them staked their entire lives on the efforts in many places, among many peoples. They and their various authority figures pored energy, manpower, and material resources into spreading the hegemony of Christianity far a wide.

Japan was no exception to this.

Another big reason was trade

Early Christian missionaries often used the prospect of privileged trade access to Japanese warlords/daimyos that allowed for preaching of Christianity in their domains. Many Japanese leaders, especially in Western Japan, were very eager for access to lucrative (and strategic) trade. A great many Japanese elites even converted, some true believers, some practical converts, and some in between.

But as the Christian communities and influence grew, soon, spreading Christianity held an economic motive for the Portuguese and Spanish.

They were more likely to get better trade deals and privileges, or at least more likely to be dealt a fair hand when dealing with fellow Christians. Trade wasn't always a simple matter of bringing money and simply exchanging it for goods. Even as the times have changed, getting ahead in business often comes down to who you know and who is willing to do you a favor.

In a land where the natives saw the Europeans as suspicious at best and dangerous at worst, being able to find a warm welcome with a native Christians or have them as middle men was a huge boon to trade and negotiation.

Christian communities also raised huge sums of money through trade and donations from the faithful. And of course, in a very religious community, the clergy had a large say in all things, including commercial activity, taxes, etc.

The last big reason was politics

Having sympathetic Christian natives made it easier for Europeans to integrate themselves into Japanese power circles, learn about the country, get the lay of the land, and obviously were a huge resource when it came time to ever negotiate for anything.

While Christian missions to Japan did not have an overtly cultural imperialist edge, it was nevertheless still advantageous for Europeans to spread Christianity there even if all they wanted were to secure lucrative and favorable trade deals.

This about sums it up.

The reasons why Japanese elites, particularly Hideyoshi (one of the famous founding fathers of Japan who also happens to be infamous for mercilessly persecuting Christians) grew to believe that spreading Christianity in the country was a sign of coming European colonialist efforts is another long, long story.

/u/AsiaExpert has covered the background, so that leaves me to go into the details! While it's quite true that the missionaries in general agreed that an armed invasion of Japan is a bad idea, that doesn't mean that individual Portuguese didn't conjure plans to invade Japan in order to achieve their religious or commercial goals.

The Portuguese entry into Japan was a great disruption of the local power balance. Wherever the Portuguese landed, the receiving daimyo, or warlord, would gain access to immense wealth and Portuguese firepower. In Japanese eyes, the missionaries were closely associated with the rest of the Portuguese commercial enterprise, and the minor daimyo sought to bring the missionaries into their lands so to attract the Portuguese "great ship" into their ports. These developments were eyed suspiciously by Toyotomi Hideyoshi, who, by the 1580s, was on the verge of uniting Japan, and increasingly put his efforts in centralizing his power.

The Jesuit fathers understood that they needed to curry favour with Hideyoshi if they were to continue proselytizing. In 1586, the Vice-Provincial of the Japan Mission Gaspar Coelho paid his respects to Hideyoshi, and the reception seemed to be excellent. Hideyoshi appeared to be frank with the Jesuit, explaining that after conquering Kyushu, he would like to invade Korea and China, set up churches there, and order the whole population to convert. For that reason, he continued, he would like the Portuguese to commit 2 carracks for the impending invasion.

Coelho, instead of distancing the Jesuit mission from local politics like his superior had ordered, agreed to Hideyoshi's request and also promised to rope in the Christian daimyo of Kyushu to help in the invasion. He also pledged the support of the Estado da India against Korea and China, which Hideyoshi did not ask for. Hideyoshi was said to be delighted at this response, but in his mind he must have been concerned about the Jesuits' influence now that Coelho had revealed his hand.

On July 24, 1587, at the urging of his Buddhist followers, Hideyoshi suddenly lashed out against the Jesuits. Here, Hideyoshi's concerns about Christianity as a decentralizing force came to a fore with his ultimatum against the Christian daimyo Takayama Ukon, to either recant his faith or face the confiscation of his fief and exile. In a perceived challenge against his authority, Takayama refused to renounce Christianity and would rather be exiled to the Phillipines, where he eventually died. In his rage, Hideyoshi ordered all Christians, foreigners and Japanese alike, out of the country.

Coelho was not happy about this sudden turn of events. In desperation, he tried to get the Christian daimyos to raise up in arms to protest against the eviction edict. When the daimyos refused, he wrote the Spanish and Portuguese authorities at Manila, Macao, and Goa to send troops to Japan in support of the Japanese Christians. However, his requests were unheeded, and he got nothing other than some weapons and a severe reprimand from his boss. Fortunately for the Jesuits, Hideyoshi never heard about this plot against him, the edict was not enforced, and Coelho himself died in 1590. The compromising weapons were sent back.

Hideyoshi's suspicions were confirmed in his mind by the San Felipe incident of 1596 (though he wasn't shogun, he was the paramount authority - known as kanpaku- next to the figurehead emperor). The Spanish ship, so named, was on its way from Manila to Mexico when it wrecked on the coast of the Japanese island of Shikoku. The crew was sent to an audience with Hideyoshi, who was toying with the idea of letting another foreign nation to trade with Japan in order to break the Portuguese monopoly. There, the pilot-major tried to impress Hideyoshi by saying the Spanish conquests of the Americas were greatly aided by the friars. This clicked with a germinating suspicion that Hideyoshi always had, and as a result 26 missionaries were crucified in Nagasaki, who became known as the Twenty-six Martyrs of Japan.

Despite all this, the Portuguese and Spanish all agree that the Japanese were all very war-like and a military invasion of the island was out of the question. This view is very unlike what some Spaniards said about China, but even in that case they have never seriously contemplated an invasion of China owing to the potentially massive logistics involved.

We've seen a lot of talk about the Japanese side of things, but not really much about the Jesuits themselves (in the 16th century, Spain and Portugal were quite linked).

Formed during the Counter Revolution in Europe, the Jesuits considered themselves a "Company of Christ." (Notice the military undertones.) And their motto was "For the Greater Glory of God.) The first Superior General, Ignatius of Loyola, was a Spaniard (born in the Basque County in Castile.) He was most definitively a papist working to increase the authority of the Catholic Church and the Pope around the world. One of the largest things Ignatius was concerned about was education and the poor quality of priests and required all Jesuits to be extremely well educated (compared to the standards of the time.) To this day, significant numbers of Jesuit priests hold advanced degrees in multiple fields (outside of Philosophy and Theology, although I would argue that if you are a priest you have the equivalent of a PhD in theology.)

As the Church's intellectuals at the time, they seemed convinced that they could increase the authority of the Church, in areas which strong governments (see China, to a large extent) and Japan, to a certain extent in the 16th and 17th centuries) by reconciling Japanese and Chinese religious traditions (e.g. the Confucian controversy) as well as spreading practical knowledge around the world. For their efforts, they were attacked by the Church yet, applauded by the Chinese and Japanese elite (e.g. the Jesuits were respected by the Kangxi Emperor in China and given a fief in Japan around the same time).

Here's where things get tricky: to finance themselves, the Jesuits relied on support from self-interested monarchies and, as such, had to involve themselves in politics (they grew to be a pretty strong force, despite various difficulties.)

So, were the Spanish and Portuguese interested in colonies in Asia? Yes. (e.g. Spanish and Portugese colonies in south east asia especially Spanish holdings in the Philippines, and Portugese holdings in Macau.) Would having a group of Christian Chinese or Japanese elites and ordinary people who were Catholic support this mission? It could not hurt. Would Spain and Portugal (in the 16th and 17th century) be able to actually colonize China or Japan? Probably not. At the time, China's economy and government, if not Japan's as well, was equal to or stronger than Spain and Portugal.