Stax, 12 bytes

ìo'½`║â↨╪U?5

Run and debug it online

It splits on a regex, and then alternately toggles case. Here's the same program, unpacked, ungolfed, and commented.

"a|A"|s split on regex /a|A/
rE  reverse and explode array to stack
W   repeat forever...
p   print top of stack with no newline
:~p print top of stack, case inverted, with no newline

Run this one