RECK is often used in poem or literature, so the utilization is very complex and sophisticated. And it is very often used in negative sentences, such as :
I reck not though I end my life to-day. --Shak.
Of me she recks not, nor my vain desire. --M. Arnold.
In this case, i think the sentence is RIGHT :
What recks he if the sky should fall?
And it means "What will/could he reck if the sky should fall?"
I can't explain the grammar here but i've found a similar example:
Then reck I not, when I have lost my life. --Chaucer.
(This sentence together with the 2 sentences above is taken from Source: Webster's Revised Unabridged Dictionary, © 1996, 1998 MICRA, Inc. They are also the examples for the word RECK.)