The ongoing experimental efforts to measure the hyperfine transition in muonic hydrogen prompt an accurate evaluation of the proton-structure effects. At the leading order in α, which is O(α⁵) in the hyperfine splitting (hfs), these effects are usually evaluated in a data-driven fashion, using the empirical information on the proton electromagnetic form factors and spin structure functions. Here we perform a first calculation based on the baryon chiral perturbation theory (BχPT). At leading orders it provides a prediction for the proton polarizability effects in hydrogen (H) and muonic hydrogen (μH). We find large cancellations among the various contributions leading to, within the uncertainties, a zero polarizability effect at leading order in the BχPT expansion. This result is in significant disagreement with the current data-driven evaluations. The small polarizability effect implies a smaller Zemach radius R_Z, if one uses the well-known experimental 1S hfs in H or the 2S hfs in μH. We, respectively, obtain R_Z(H) = 1.010(9) fm, R_Z(μH) = 1.040 (33) fm. The total proton-structure effect to the hfs at O(α⁵) is then consistent with previous evaluations; the discrepancy in the polarizability is compensated by the smaller Zemach radius. Our recommended value for the 1S hfs in μH is 182.640(18)meV.