I think it's possible that the term "STEM" obfuscates the problem. I don't believe believe there are any shortages in the maths and sciences. There's not a lot of science graduates in the school I graduated from to be sure, but there wasn't exactly a lot of demand for them either. Engineering is a wash -- it really depends on the discipline. Comp Sci guys though are in high demand -- basically every CS grad from my uni had job offers before they graduated, and it's not a prestigious school or hi-tech hotspot.
The only people who believe theres a shortage of 'STEM' workers are the guys that don't work there. Sure there are 'STEM' fields where you can more easily get a job than other industries ie CompSci vs Renaissance Studies. But its nowhere near the golden fields of opportunity fallow of workers the politicians and ceos like to whine about. Even CompSci has severe challenges of outsourcing, overwork, and stability while many other disciplines like BioSci are overcrowded with Medschool washouts and is 50% forsaken grad students/postdocs toiling on the plantation. We could argue that we need science till the cows come home but the fact remains that society does a lot to pump up STEM at school but virtually abandons students by the time they graduate. At least with Humanities you don't have the talking heads pushing disinterested women, minorities, and h1bs into an already saturated market at the same time.