That's an interesting one! (Spoiler: The answer is 'No.') Hadn't heard about that, even if I'm aware that some English protestants (and others) were fans of Sweden during the Thirty Years War period and 17th century. (e.g. The Svvedish discipline, religious, civile, and military)

As I wrote a bit in a longer post recently, Sweden's break with Catholicism was not a sudden one; Lutheranism (as it'd later be known) started to catch on before it was an actual schism and was introduced gradually starting in the 1520s. The Church of Sweden broke with Canon Law in the 1530s but it wasn't until 1593 that the Augsburg Confession officially became the sole accepted doctrine, to the exclusion not only of Catholicism but also Calvinism and Zwingli-ism. (Orthodox Christianity was tolerated, but only because it was a concern of the small Russian minority off at the far end of the Gulf of Finland. Converting them was put on a to-do list for the government. In the 17th century a Russian translation of Luther's Small Catechism was printed and disseminated but never gained much traction)

Each step away from Rome had caused more staunch Catholics among the clergy to quit or more often go into exile; but there was significant doctrinal variance in the mid 1500s. The church ordinance of 1571 largely ended that, and established a standard and non-catholic ceremonies. The adoption of the confession in 1593 was to an extent a restoration and reconfirmation of that, as the king Johan III had tried a rapprochement with the Catholic church while retaining much of Protestantism, a move largely opposed by the Swedish clergy. (there was not nearly the back-and-forth of England) So by 1571 you could not simply not practice the catholic liturgy in Sweden legally anymore. There were certainly some crypto-Catholics around but no catholic rites were performed publicly and the church had no presence. Nobody in the country was allowed to practice Catholicism, apart from foreigners granted exemptions.

Tolerance continued to decrease; taking aim at the risk of Swedes studying abroad having their faith undermined; abandoning the faith was punishable by exile and loss of inheritance rights. The Örebro statues of 1617 further toughend the rules, banning Swedes from having any contact with any servant of the Polish government and attending Jesuit or other 'papist' colleges abroad. Abandoning the faith was now treated as treason, and thus punishable by death, as was proselytization of Catholicism.

In practice, death sentences were rarely carried out. People were let off if they recanted their conversion to Catholicism, so only a half-dozen Swedes were executed for converting to Catholicism in the 17th century, and 1-3 Sami who'd practiced their native religion. Very small numbers compared to the 400 or so killed in the witch hunts of the same century.

In other words, there was no castration going on by the early 1600s. Nor any Catholic priests around to castrate. As said, Swedes who converted would generally be forced to recant and (as I'll get to) foreigners expelled, so it all seems rather bizarre. Not really clear how it'd work as an effective punishment either? I mean being castrated is unpleasant but wouldn't really stop them from being catholic priests.. or even stop them from doing anything except something they weren't allowed to do anyway (in re: vows of celibacy). Which seems to be the actual point; a bit of Protestant "trolling" along the lines of "If you're really celibate you shouldn't have a problem with this, and if you do you're just proving papists are hypocrites and fornicators!"

Anyway in Sweden, tolerance would be expanded somewhat a few decades later, when catholic diplomats would be allowed to practice their faith in embassy chapels. It's likely foreign immigrants partook in these services despite being banned from it but the general population did not (or risk the death penalty). The Jesuit Johannes Sterck from Aachen (source, German) began a a legation priest in Stockholm but attempted to start a mission and was sentenced to death in the 1670s, but his sentence was commuted to banishment. There does not seem to be any cases of foreign priests being executed.

Since the claim seems to originate from this anonymous pamphlet from 1700, attributed to Daniel Defoe, there's no reason to look beyond the 16-17th century. I suppose the problem with pointing to Sweden as an example for how to deal with the 'growth of papism' is that Sweden simply didn't have any papists, except for a handful of foreigners (who - René Descartes for instance - were treated with great suspicion). There'd been zero tolerance for over a century. Attending church services was mandatory. From the start of the 1600s Sweden instituted a conscription system, and in the military religious rites were daily and mandatory. Starting in 1685 learning to read and know biblical texts and Luthers Small Catechism was mandatory for the entire population, and tested annually. It'd been translated to Swedish and Finnish almost as soon as it'd been written, but in the 17th century it was translated to the kingdom's minority languages too. Besides the aforementioned Russian, it was also translated into Sami, Estonian (both Tallinn and Tartu dialects), Latvian and even the Lenape language of the Indians living near the New Sweden colony in America. It was a devoutly Lutheran country in that era.

(Sources same as earlier; mainly Carl Cornelius for the finer details of church history)