Whenever you are confused by some wierd perl syntax alway run to the wonderfull B::Deparse module.

If you use it with the -p option to add extra parentheses it will often (almost always) be clear what is going on.

Using it on your example I get:

perl -MO=Deparse,-p -e '$x=1;print 1 if $x or print 0'
($x = 1);
(($x || print(0)) and print(1));
-e syntax OK
[download]

It should be clear that all the explanations concerning 'or' and expressions are true.

Your other question is related.
The interesting part is the line &one if $x or &zero;B::Deparse deparses it as (($x || &zero) and &one); $x is false, so it must evaluate &zero.
&zero prints 'zero' and returns false.
So the entire expression is false and &one is not called.

To sum it all up: use B::Deparse and all will be revealed.