Welcome to the forum!
No, a degree is not mandatory for an H-1B - I myself am proof of that
I got an H-1B based solely on work experience and with 10 or more years the door should be open to your husband on that route.
However if your hubby's doing an international transfer (i.e. he currently is employed by company based both in the US and elsewhere) then it should be even easier - via the L-1A or L-1B visa which is not regulated in the same way.
The Visa's both have a slightly short life-span but they can be both renewed and also used as a springboard for obtaining a green-card followed by naturalization. As a spouse you would be on an L2 Visa which, after obtaining an Employment Authorization Document (EAD) you'd be able to use to obtaining work at any other company.
Hope this helps!