We present a detailed analysis of stellar rotation along the main sequences of NGC 1866 and NGC 1856, two young (~200-300Myr) massive clusters in the Large Magellanic Cloud, using MUSE integral field spectroscopy. Differences in stellar rotation have been proposed as an explanation for the extended main sequence turn-offs and split main sequences in these clusters. In agreement with this idea, we find strong links between the photometric colours of the cluster stars and their projected rotation velocities, Vsini. While stars blueward of the split main sequences are characterized by a range of relatively low spins, those with redder colours are fast rotators. Following a statistical correction for inclination, we measure mean equatorial velocities for the red main-sequence stars in both clusters of Veq=300km/s, corresponding to 70-80% of the critical values predicted for such stars by current stellar models. We discuss these findings in the context of the different scenarios proposed to explain the stellar rotation distributions of young massive clusters. We further investigate whether the high rotation rates provide a natural explanation for the high fractions of Be stars we observe in both clusters, peaking at >=50% at the turn-off. We argue that if ~85% of the critical rotation rate is high enough to trigger the formation of a decretion disk, most upper main sequence stars in the clusters are expected to become Be stars before leaving the main sequence.