The daf a few days ago (Shabbos 12) discussed ביקור חולים on Shabbos. The gemara says that someone who goes to visit a sick person on Shabbos should say שבת היא מלזעוק ורפואה קרובה לבא to the sick person. Rashi explains, we are telling the sick person to try not to be sad because it is shabbos and a person is supposed to be happy on shabbos. The Ran has a different peshat. He says we are telling the sick person that since it is shabbos we can't daven for him.
Rashi's peshat fits in better with the מי שבירך לחולים, we say the מי
שבירך and then we tell the people in shul, don't be upset about the sick
person it is shabbos. However, according to the Ran it is a bit
difficult, we are contradicting ourselves, we just said a prayer for the
sick person and we end off by saying we really aren't allowed to daven
for you. The fact is that the whole idea of making the מי שבירך לחולים specifically on Shabbos is very strange. Shabbos is the one day when we aren't supposed to be sad and daven for sick people and yet we specifically say it then.