Can high blood sugars be the cause of my baby's eczema rash?

No. I don't know of any evidence of a link between blood glucose levels and eczema. The good news is that most babies outgrow infantile eczema; the bad news is we can't tell you when - could be a week, could be a year or more.
Dehydrated? Eczema is exacerbated by dry skin. High blood sugars can cause dehydration which would worsen eczema. To my knowledge there is no direct correlation between eczema and blood sugar concentration alone.
Blood sugar/eczema. British journal of dermatology, august 1925 (37) 364-370 "for many years a vague connection has been recognized between derangements of carbohydrate metabolism and the development of dermatitis of various kinds" thus, there may be a connection.