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It seems to be a result of J-M Bony that every nonnegative function in $C^{2m}$ is a sum of squares of two $C^m$ functions, which means that every $C^\infty$ function is the sum of squares of two $C^\m$ functions for any $m$ (which, I suppose, does not mean that you can do it with two $C^\infty$ functions -- the counterexamples are attributed to Paul Cohen and D.B.A. Epstein -- see references 1and 4 in the cited paper): the reference is:

Bony, Jean-Michel(F-POLY-CMT) Sommes de carrés de fonctions dérivables. (French. English, French summary) [Sums of squares of derivable functions] Bull. Soc. Math. France 133 (2005), no. 4, 619–639.

For functions from $\mathbb{R}^k \rightarrow $\mathbb{R}_+$ these results are extended in:

Nonnegative functions as squares or sums of squares􏰿 Jean-Michel Bonya, Fabrizio Brogliab, Ferruccio Colombinib, Ludovico Pernazzac (J. Func. An, 2006)