Expressing the entire business in terms of SphericalBesselJ[] cures the problem:
NIntegrate[r With[{x = r BesselJZero[15/2, 1]},
Sqrt[2 x/π] SphericalBesselJ[7, x]]^2, {r, 0, 1/50}]
1.1879560281974252*^-27
The nice thing about SphericalBesselJ[] is that it does not auto-evaluate to a potentially numerically unstable combination of trigonometric functions, unlike BesselJ[15/2, x].