Probably not.
MINDEF has no problems with reservists who happen to be working /
studying overseas to serve part-time with foreign armed forces (for
example, a friend of mine who was studying in the UK served as a
reservist with the Royal Green Jackets - he trained once a week,
two weekends a month and two 14-day field camps a year).
But if you joined the Foreign Legion, that would of course be a
full time thing, meaning that you would be going to France
specifically to join La
Legion, rather than it being incidental to any overseas
study or overseas work. Among other issues, that would be, in the
eyes of MINDEF, an unjustified interference with your reservist
liability in Singapore.
Of course, what the picture is if you've already completed your
reservist cycle - that I'm not sure.
In any event, if someone is hell-bent on joining the Legion, what
can Singapore do to stop him? He'll be in France, beyond the reach
of Singaporean authorities. He won't be able to return to
Singapore, but heck, after he gets French citizenship, he may not
want to, anyway.
The problem, of course, arises where he, for whatever reason,
doesn't manage to complete his tour of duty with the Legion - in
those circumstances, he won't have French citizenship, and he'll
probably be arrested in Singapore if he comes back here. His
recourse may then be to apply for political asylum in a European
Union state. Not sure what the odds of his success would be,
though.