- This is, of course, the same as saying about spacings between uniform points on a segment (you can fix one of the points on the circle). Let it be the segment $[0,1]$.
Now the joint distribution of $I_1,\dots, I_{n}$ is the same as of $E_1/E,\dots, E_n/E$, where $E_1,\dots, E_n$ are iid exponential distributed, $E=\sum_{k=1}^n E_k$ (see Devroye Non-Uniform Random Variate Generation, p.208). So the distribution of $I_{(1)},\dots, I_{(n)}$ is the same as of $E_{(1)}/E,\dots, E_{(n)}/E$. But the joint distribution of $\{E_{(k)}-E_{(k-1)},k=1,\dots,n\}$ ($E_{(0)}:=0$) is the same as of $\{(n-k+1)^{-1} E_k,k=1,\dots,n\}$ (ibid, p.211).
So the distribution of $J_1,\dots, J_n$ is the same as of $\{(n+k-1)^{-1} E_k/E,k=1,\dots,n\}$, where $E_1,\dots, E_k$ are iid exponential rv's, $E=\sum_{k=1}^n E_k$.
From here you can, of course, find the distribution of maximum and minimum (but nothing very pleasant there, as the variables in question are not identically distributed; a formula for the expectation looks extremely ugly).