You could think of integers as pairs of natural numbers $(a,b)$ modulo the equivalence $$(a,b) \sim (c,d)\quad \mathrm{if}\quad a+d = b+c.$$ (In other words, you think of $(a,b)$ secretely as $a-b$.) Addition is just entrywise addition of natural numbers: $$(a,b)+(c,d) = (a+c,b+d).$$ Negative numbers are numbers of the form $(0,a)$. Now the equation $-7 -3 = -10$ is $$(0,7) + (0,3) = (0,10).$$ In summary, you can restrict yourself to adding natural numbers, provided that you consider pairs.

I'm not sure this helps in your friend's case, though.

You could think of integers as pairs of natural numbers $(a,b)$ modulo the equivalence $$(a,b) \sim (c,d)\quad \mathrm{if}\quad a+d = b+c.$$ (In other words, you think of $(a,b)$ secretely as $a-b$.) Addition is just entrywise addition of natural numbers: $$(a,b)+(c,d) = (a+c,b+d).$$ Negative numbers are numbers of the form $(0,a)$. Now the equation $-7 -3 = -10$ is $$(0,7) + (0,3) = (0,10).$$ In summary, you can restrict yourself to adding natural numbers, provided that you consider pairs.

I'm not sure this helps in your friend's case, though.

Edit

Negative numbers do not arise in nature, of course. It is the result of comparison. The above set up with pairs is precisely that. $(a,b)$ is positive if $a>b$, negative if $a<b$ and zero if $a=b$. This could perhaps be woven into some sort of narrative for your friend.