Is there a correlation between red gums & autoimmune diseases?

Not directly: Red gums typically involves some level of active infection process that certainly activated the immune system in that individual to a greater level. That condition may exasperate an underlying autoimmune disorder adversely. It certainly does not directly cause the autoimmune disorder condition directly but it certainly does aggravate it.

Lichen planus: Lichen planus, an autoimmune disease may give you a reddish gum with lattice work of white patches... Mouth may feel dry. Gum may be glossy and inflame... You cannot treat lichen planus, only palliatively to provide comfort... (drug such as coricosteroid, antiinflammatory).