The emission of neutron pairs from the neutron-rich N=12 isotones ^{18}C and ^{20}O has been studied by high-energy nucleon knockout from ^{19}N and ^{21}O secondary beams, populating unbound states of the two isotones up to 15 MeV above their two-neutron emission thresholds. The analysis of triple fragment-n-n correlations shows that the decay ^{19}N(-1p)^{18}C^{*}→^{16}C+n+n is clearly dominated by direct pair emission. The two-neutron correlation strength, the largest ever observed, suggests the predominance of a ^{14}C core surrounded by four valence neutrons arranged in strongly correlated pairs. On the other hand, a significant competition of a sequential branch is found in the decay ^{21}O(-1n)^{20}O^{*}→^{18}O+n+n, attributed to its formation through the knockout of a deeply bound neutron that breaks the ^{16}O core and reduces the number of pairs.