ושבעה עשר בתמוז הפך לנו לשמחה ולששון*
The Brisker Rav said in the name of his father R. Chaim, "when Chazal established the four fasts, It was not that they established the fasts and then exempted the חולה (sick) from fasting.
They established the fasts specifically only for healthy people and not for the sick.
Therefore, there is no point for a חולה to be Machmir in fasting. If one does go ahead and fasts it is not considered participating in the fast".
Some Poskim** add that according to R.Chaim if חולים are part of and are needed to complete the Minyan then you might not be allowed to Lain
ויחל since there is no Minyan of Yidden who are participating in the fast even if all ten are fasting.
Mori V'rabi Harav Hagaon R. Shlomo Miller Shlita said in jest that perhaps we can add one more Brisker Chumra, "not to give an Aliya on a fast day to anyone who might be a חולה."
סליחות לי"ז בתמוז*
תשובות והנהגות ח"ז להגר"מ שטרנבוך שליט"א**