Since the answer is not the same as $EGG$, $E$ is not $1$. However, $E \times E$ is less than $10$, as the answer has only three digits, so $E$ is $2$ or $3$.
The units column must carry, otherwise it would be the same as the tens column, and $A$ and $M$ are different. But this means the tens column must also carry.
But, if $E=3$, then this carry means $J = 10$, which cannot happen. Therefore $E=2$.
Since the tens column carries, $J=5$. For the carries in the tens and units columns, $G$ must be at least $5$. Trying each of these combinations leaves two of the letters having the same value, unless $G=8$. This gives $E=2,G=8,J=5,A=7,M=6$, with $288 \times 2 = 576$.
This problem is taken from the UKMT Mathematical Challenges.