= Solution
The union area is a bounded measurable function of the centres, with $0\le X\le n\pi$. Fix all centres except one, and let $B$ be the union of the fixed disks. Replacing the remaining disk $D$ changes the area to $|B|+|D\setminus B|$. The second term always lies in $[0,\pi]$, so the area changes by at most $\pi$, not merely by the weaker bound $2\pi$.
Thus $X/\pi$ has coordinate differences at most one for the $n$ independent centres. The preceding <martingale> bound gives
$$
\boxed{P(|X-EX|\ge t)\le2\exp\left(-\frac{t^2}{2\pi^2n}\right)}.
$$
One may take $c=1/(2\pi^2)$, independently of all centre distributions. Boundedness of the area ensures the conditional expectations exist even when the centre locations have no finite moments.
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