Recovering 3 private keys if Eve knows that the keys are shared prime numbers and knows their public keys, How would this be done?
Yes, your thinking is correct! Using a "shared prime" when calculating $n_A$ and $n_B$ is disastrous for security simply through the fact that it reveals $p$, $q$, and $r$ with a few calls to the $gcd$ algorithm. 

Luckily this incredibly unlikely to happen with the use of good randomness. In the case of 512-bit RSA, there are roughly $2^{503}$ 512-bit primes which, when using the birthday bound, equates to about $\sqrt[2][2*2^{503}log(1/(1-0.5)] \approx 6.02520 * 10^{75}$ generated primes needed to achieve a 50% probability of a collision.

Source: http://www.loyalty.org/~schoen/rsa/