I would make your interrupt count up to 60 in some variable providing some 'count-enabled' flag was set, and back to zero again when it overflows, it can do that in the background, then as soon as the foreground task detects a switch closure (your push-button) it clears the 'count enabled' flag so stopping the incrementing and as soon as you release the push-button, the code sets the count-enabled flag again.
init:
count_enabled = 1
SETINT ...
start:
IF push_button = closed THEN
count_enabled = 0
ELSE
count_enabled = 1
ENDIF
GOTO start
interrupt:
SETINT ... renable your interrupts ' it depends how you got here, e.g. timer time-out, or input transition
IF count_enabled = 1 then
seconds = seconds + 1
IF seconds > 60 THEN
seconds = 0
ENDIF
ENDIF
RETURN