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.][1] 

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)][2]


  [1]: http://www.math.polytechnique.fr/~bony/S2KR_bullSMF.pdf
  [2]: http://www.math.polytechnique.fr/~bony/S2KR_bullSMF.pdf