FWIW, considering that I have never tried to do a front squat, but looking at online front squat instructions and your description, when I try to arrange my arms like this with a broom handle to stand in for the bar, I run into the same problems you describe, so whatever the problem is, I don't think you are particularly freaky.

That said, I found a positioning explanation for learning front squats that says for the bar position you should do a "Frankenstein squat" first, i.e. keep the arms outstretched with a bar supported by the shoulders, and that that would be the same bar position as in the front squat, and that instruction said that the shoulders should be forward and the bar resting against the throat, but the thoracic spine should not be rounded. I can balance my broom handle on my shoulders in that position, but when I then bring my hands back I can't bend my wrist far enough to get my fingers beneath it, to match the pictures of front squat positions I've seen, even if I don't want to grip.

Sorry I can't give advice how to resolve the problem, but you are definitely not alone.

ETA: I've seen instructions that explain a front squat variety with crossed arms, as an alternative for people who lack wrist flexibility/have trouble with the wrists in the standard "clean grip" variety:
http://breakingmuscle.com/strength-conditioning/a-primer-on-front-and-back-squats-crossed-arm-clean-grip-low-bar-and-high-bar
The disadvantage is apparently that the bar rests less secure, but maybe that variant might work better for you?