Can eczema cause eyes to be swollen shut after playing with a dog?

Yes.  Please trust your instincts because you're on to something.  Has the child had blood tests for dog performed (a Blood RAST test will rate his allergy level from 0 to 100, with 100 being the highest).  A skin test is neither reliable nor can it accurately measure the severity of the allergy (important measurements to get annually to determine if allergies are going away or getting worse).  Your child should also be tested for PEANUT, TREE NUTS, FISH, SOY, WHEAT, EGG, and MILK as he may not have an actual allergy to DOG, but to these foods which are frequently found in commercially-sold dog foods and treats.    Oh yeah, and if your dog chews on dog toys made from natural rubber (latex) your son could be allergic to latex, too.

I went through FIVE pediatricians with a my child's eczema (sometimes crusty and bleeding).  All of them pooh-poohed the eczema saying it would go away at the same time they prescribed strong steroid creams and prescription antibiotic creams to prevent impetigo.  But it continued to ebb and flare up unpredictably. 

I have since found out that over one third of eczema cases in young children is caused and can be traced to bad allergies to protein-rich foods, animal dander, pollen and molds.  Turns out my eczema-baby went into anaphylactic shock at 16 months from peanut dust stirred when wrapped candy was poured into a bowl two feet from him.  It was then that the pediatrician took it seriously. His subsequent blood tests showed he was severely allergic to several hard food proteins, and had a mild allergy to dog (so mild was the blood RAST rating that we just kept him away from our little dogs and he's now outgrown the allergy.  Turns out it was a contact allergy, so keeping dogs away was the solution.  And getting rid of real rubber toys and making sure dog foods had no peanuts (have to call manufacturer as no federal laws mandating listing of food allergens on pet food.

  Kids die from anaphylaxis.  Please find an allergist that will take you seriously and administer ALL of the proper blood tests.  Then, if any show a strong allergy, your new allergist or pediatrican can prescribe EPI-PENS which you can carry at all times and you can use them to save your kid's life.  God bless and good luck.  Your child is lucky to have such an observant parent.  Again, trust your gut instinct, it's better than a medical diploma.

  By the way, the wonderful allergist I found recommended a pediatric nutritionist who recommended lots of pure olive oil in my kid's diet.  I drizzled it in baby food, milk bottles and later in mashed potatoes, etc.  Why such a keen interest in pure Italian olive oil?  Well, it cured my baby's eczema completely.

Signed,

Been there, done that.