Your question regarding the plague, I assume, is not Mercutio calling a a plague down on the Capulet and Montague families when he is fatally wounded by Tybalt in Shakespeare's Romeo and Juliet.
I am hurt.
A plague a' both your houses! I am sped. (III.i.90-91)
The plague, however, does play a prominent "part" in the play. When Friar Lawrence sends Friar John to Mantua with a note for Romeo apprising Romeo of the plans the friar and Juliet have made to avoid her marriage to Paris, Friar John is unable to leave the city. In Act Five, scene two, Friar Lawrence receives the news from Friar John that the letter for Romeo was never delivered, and he explains why.
Going to find a barefoot brother out,
One of our order, to associate me
Here in this city visiting the sick,
And finding him, the searchers of the town,
Suspecting that we both were in a house
Where the infectious pestilence did reign,
Seal'd up the doors, and would not let us forth,
So that my speed to Mantua there was stay'd. (V.ii.5-12)
This is not a detail lightly placed within the play. Everyone in England, in Europe, realized the importance of containing the plague. One of the most social gathering places in England was the theater, and when there was any risk of plague, the theater was the first place to be closed because it gathered people from all socioeconomic groups, from all over the kingdom.
The plague knew no barriers based on age, gender or social prominence. The most popular was the bubonic plague, though there were three different kinds. It moved swiftly, killing most commonly within a week. Some people did survive, but a majority did not; it wiped out entire towns. It was known as the Black Death, and the fact that Friar John was quarantined is no surprise whatsoever. Houses were boarded up; no one was allowed in or out until contagion could absolutely be ruled out.
The Italian author, Giovanni Boccaccio, wrote that the plague so terrified people that family members would abandon one another. The greatest calamity was that parents would abandon their own children:
What is even worse and nearly incredible is that fathers and mothers refused to see and tend their children, as if they had not been theirs.
With this information in mind, there can be little doubt as to why Friar John would not have been allowed to leave Verona and travel into Mantua, or anywhere else. No amount of pleading would have helped. To let one infected person through could mean the death of hundred or thousands. For instance, the Black Death was...
...the first great pandemic of recorded history, with death rates reaching and in places exceeding 70 percent.
It is this detail, fear of the plague, on which the plot of Romeo and Juliet pivots. Had Friar John made it through, it is safe to assume that Romeo and Juliet would have been reunited.