I have a 2001 6 cyl Camry and I broke one of the three studs that hold the housing. How do you get it out?

 

all three is bad luck
broken because of corrosion

if the engine block side of that housing for the thermostat cover
can not be removed or purchased

two options

if a bit of the stud is workable
heat up or weld a nut to what is viable
then undo

or loc-tight two nuts to remaining thread to undo carefully

if broken at the base of the stud
drill them out straight ... tap a new thread and helicoil kit
three new studs