There are two accounts written in the 500s, one contemporary to the supposed events, one later. It is not automatic that a later account means false; it does mean we have something to cross-reference to work out if there are any common elements. Unfortunately, due to some other circumstantial factors, it is unlikely either version is true.

The first version is by Procopius of Caesarea who was contemporary to the alleged events; I've quoted a portion, although if you want to read the whole thing I will put it as a reply to this answer.

About the same time certain monks arrived from the (country of the) Indians, and learning that the Emperor Justinian had it much at heart that the Romans should no longer buy silk from the Persians, they came to the king and promised that the Romans should not have to purchase the article either from the Persians or from any other nation; for they had lived, they said, a long time in the country where there were many nations of the Indians, and which goes by the name SERINDA.

It is unclear where Serinda is actually referring to, as various locations have been named; it may be farther west than contemporary India, but it definitely is not referring to China. Sogdia (a location in ancient Iran) is a popular choice; the problem is while we have Sogdian silks from the 7th and 8th centuries, we don't have any from the 6th century; the only evidence in that direction is in a Sogdian embassy in Constantinople having silk for sale in 568, but the silk is not mentioned as Sogdian and could easily have been imported.

The account goes on to explain that the monks explain that bringing caterpillars as a way to bypass the Persian trade would not work, but eggs could be covered with dung and be hatched upon arrival. Justinian offers "large promises of reward" for doing the task, and the monks "went back again to India" and return with the eggs in question, successfully nurturing them into caterpillars and henceforth a Byzantine silk industry.

There is no mention of the theft being discovered; maybe we'd expect that from an Indian source, but none of them mention this, possibly for the reason the story might be utter nonsense. I'll get to why in a moment, but first let's look at the other account, from Theophanes of Byzantium, who was writing at the end of the century.

Now in the reign of Justinian a certain Persian exhibited in Byzantium the mode in which (silk) worms arc hatched, a thing which the Romans had never known before. The Persian on coming away from the country of the Seres had taken with him the eggs of these worms (concealed) in a walking stick, and succeeded in bringing them safely to Byzantium.

In the first story, Justinian requested the silk heist be performed, and the monks did the deed and came back. In this variation, a "certain Persian" did the deed, and just brought it on their own. Additionally, the Persian came from "the country of the Seres".

"Seres" is its own loaded word, sometimes used to refer to India, sometimes used to refer to China (depending on the historian) but the important thing is how the Byzantines of the 6th century understood the word, which seems to not reference a specific place at all: just "the place where silk comes from". (Notice that to get the original "two monks in China" story you have to mash the two stories together and presume that "where silk comes from" is meaning China.)

The fact is that sericulture is a complex process which requires an established industry, and is not as simple as one or two people bringing a set of eggs. Mulberries need proper cultivation, there are many ways silkworms can get infected, and the process of reeling the silk itself is also delicate.

We have records from China -- which carefully noted when sericulture occurred outside the country -- that the late Han dynasty (25 to 220 CE) mentions Daqin as having a silk industry. In the Tang dynasty (618-906) this region was called Fulin.

It gets referenced once as a place with animals of

...the donkey, the mule, the camel, and the mulberry silkworm.

Where is Daqin/Fulin? This is a controversial and complex debate, but there is strong reason to think it was in reference to Syria, and that sericulture started there at least by the 5th century.

This means sericulture likely simply have come from there via a more gradual transfer, which is far more plausible than assuming an entire industry -- which requires more than just access to eggs -- was brought in via walking stick.

..

Jacoby, D. (2004). Silk economics and cross-cultural artistic interaction: Byzantium, the Muslim world, and the Christian west. Dumbarton Oaks Papers, 58, 197-240.

Muthesius, A. (1993). The Byzantine silk industry: Lopez and beyond. Journal of Medieval History, 19(1-2), 1-67.