Have you tried not adding egg?
In my experiments, egg is needed when you add breadcrumbs, but if you just form some ground beef into a patty and fry it, it doesn't fall apart.
Hm, let me clarify that.
If you take a single blob of ground beef straight from the styrofoam and press it in a patty press, it doesn't fall apart.
If you take two pieces and put them next to each other in the patty press, it does fall apart as you cook it.
So when I need to add an extra bit, I add it on top.
That works.
So there is at least one way you can make a hamburger without adding egg.
Now, if you wanted to mix spices into the beef, then you might have a problem.
I'll admit that.
Better than nothing, though.
cooking.stackexchange.com/users/304/Kate Gregory
A bit of corn starch would help if you cannot add egg at all.
cooking.stackexchange.com/users/7783/K2so
I've never used anything other than beef, dry spices, and sometimes garlic to make my hamburgers.
Truthfully, I much prefer the taste of a burger with no binders.
We buy our beef by the half cow, and make plenty of burgers.
Freezer -> defrost -> Mix beef with spices ->
Form burgers while still cold (I make a round ball with the meat, them compress it)
-> straight to grill.
We've never had any issues.
I will also say, that my wife is much better at making evenly sized, well formed burgers, that hold together really well..
cooking.stackexchange.com/users/231/talon8
There are two things you need to do to make perfect hamburgers without egg or other binder.
First knead the meat balls so that the meat binds together.
Kneading give you a better bind you will get.
Press the meat firmly into your mould before popping out.
Second, after forming into shape, refrigerate the meat pats.
This process first stretches the proteins in the meat.
When chilled, the proteins contract again but now they are twisted together in a bond.
When cooking, the pat will hold together just a well as any that you buy in a shop.
If find that this works even when I add liquids to the meat.
I often season my meat with Worcester sauce mixed in.
Even with this extra liquid the meat will hold together just fine.
No egg required.
cooking.stackexchange.com/users/6615/Rincewind42
You can use what is called a flax egg , it is 1:3::milled flax seed: water (4Tbsp is about 1 egg in terms of binding strength).
Not only is flax the secret glue in many vegan recipes, but it might be a good additive for your wife's diet.
cooking.stackexchange.com/users/15/mfg
You can fry it without an egg at all.
I just form a patties, leave them on the table or in the fridge for a hour or so and then fry it.
Nothing fall apart, everything is fine.
cooking.stackexchange.com/users/7845/Sergey Larionov
I too had an intolerance to egg when I was pregnant with our last child.
In place of egg when making hamburgers or meatloaf I used a jar of veal baby food.
It helped bind and keep the dish moist.
Doesn't really add or subtract from the flavor of the beef either.
Best of luck to you!
cooking.stackexchange.com/users/20882/Genia
Time for some molecular gastronomy.
Transglutaminase ("meat glue") should work perfectly.
http://en.wikipedia.org/wiki/Transglutaminase
cooking.stackexchange.com/users/7036/Michael
1
Tbs flour for 1 Pound of meat
cooking.stackexchange.com/users/20134/V Groves
Using pork mince will make a good burger as the proteins in pork meat bind naturally
and so you do not need the egg to bind the mix.
Just work the meat a little to encourage a good mix.
cooking.stackexchange.com/users/42932/meatmixer
My wife is having pregnancy cravings for (homemade) hamburgers, but has developed a prenatal allergy / intolerance to egg.  What is a possible replacement for egg to stop a hamburger pattie from collapsing into mince during cooking?
Substitution for egg to bind hamburger mince
cooking.stackexchange.com
<substitutions><eggs><hamburgers>
cooking.stackexchange.com/questions/18607
cooking.stackexchange.com/users/2902/johnc
Try making the patties without binders, or using a flax egg, a jar of veal baby food or a bit of corn starch. The proteins in pork mince will bind naturally, making it a good choice for burger meat. You don't necessarily need a binding agent. You can also try using a flax egg or a bit of corn starch.
You don't need to add a binder like egg. Pork mince will hold together on its own. Try using a flax egg or adding a bit of corn starch.