You can sanitize your hands only to a certain degree, so skin contact with chilled wort or beer should be avoided unless you pasteurize (the wort) again, afterward. Maybe wear gloves when squeezing?
On the other hand, there's really no need to strain/filter the (chilled) wort that tightly (finely). Just by letting the chilled wort sit in the (covered) pot for 30 minutes, most trub will have precipitated. Pour or siphon the clear wort on top carefully into your fermenter, leaving the trub (with some wort) behind. Even some trub in your fermenter won't harm your beer. It may be even helpful, and it all precipitates out after fermentation has finished, while it conditions, before packaging (bottling/kegging). Cold crashing generally speeds up precipitation.
That leftover trubby wort can be strained or refined but if touched by skin or other non-sanitary devices, should be re-pasteurized before adding to your batch in the fermenter. I often do that under the paradigm: "No wort left behind!"
From a 5 gallon batch I can reclaim about 2-3 quarts from the trub by straining through tight mesh nylon "hop bags." The reclaimed wort gets then re-pasteurized at 150-160F for 20' or simply reboiled. It can then be added to the main batch (if volume is lowish), fermented on the side in it's own (small) fermenter, used to make syrups, or (yeast) starter wort.