The UUID specification details several "versions" which are methods for generating the UUID. Most are aimed at ensuring uniqueness (that's the main point of UUID) by using, e.g., the current date. This is efficient but means that while the generated UUID are unique, they are also predictable, which makes them inadequate for some security usages.
The "version 4" UUID generation method (in section 4.4), however, is supposed to use a cryptographically strong random number generator. 6 of the 128 bits are fixed to a conventional value (to indicate that this is a version 4 UUID, namely), so this leaves 122 bits from the RNG.
If the underlying RNG is secure (e.g.
/dev/urandom on a Linux/MacOS/*BSD system, or
CryptGenRandom() on Windows) then given many generated UUID, an attacker is not supposed to be able to predict the next one with success probability higher than 2-122, which is adequately small for most purposes, including launch codes for nuclear missiles.
122 random bits ensure uniqueness with high probability. If you generate many version 4 UUID and accumulate them, you may expect to encounter your first collision after about 261 UUID -- that's about 2 billions of billions; simply storing that number of UUID would use more than 30 millions of terabytes. If you consider "only" 1012 such UUID (one thousand of billions, storable over 16 terabytes), then risks of having two identical UUID among these are about 9.4*10-14, i.e. about 700 thousands times less probable than winning millions of dollars at the lottery.
Therefore, UUID are appropriate for security purposes if (and only if) they are "version 4" UUID generated with a secure RNG.