We report results from a deep search for redshifted HI 21cm absorption in 55 strong MgII$\lambda$2796 absorbers (having $W (MgII) > 0.5 \AA$) at intermediate redshifts, $0.58 < z_{\rm abs} < 1.70$, with the Green Bank Telescope (GBT) and the Giant Metrewave Radio Telescope (GMRT). Nine detections of HI 21cm absorption were obtained, all at $1.17 < z_{\rm abs} < 1.68$, including three systems reported earlier by Gupta et al. (2007). Absorption was not detected at $> 3\sigma$ significance in 32 other MgII absorbers, with 26 of these providing strong upper limits to the HI 21cm optical depth, $\tau_{3\sigma} < 0.013$ per $\sim 10$ km/s. For the latter 26 systems, the spin temperature $T_s$ of the absorber must be $> [800 \times f]$ K (where $f$ is the covering factor), if the HI column density is $\ge 2 \times 10^{20}$ cm$^{-2}$, i.e. if the absorber is a damped Lyman-$\alpha$ system (DLA). Data on the remaining 13 systems of the sample were affected by radio frequency interference and were hence not useful. Excluding "associated" systems (within 3000 km/s of the quasar redshift), the detection rate of HI 21cm absorption in strong MgII absorbers is $x_{\rm 21,MgII} ({\bar z} = 1.1) = 25^{+11}_{-8}$%, at a $3\sigma$ optical depth sensitivity of $\sim 0.013$ per 10 km/s. Comparing the detection rates of HI 21cm and damped Lyman-$\alpha$ absorption in strong MgII absorber samples yields a detection rate of HI 21cm absorption in DLAs of $x_{\rm 21,DLA} ({\bar z} = 1.1) = (73 \pm 27)$%, consistent with the detection rate of HI 21cm absorption in low-$z$ DLAs. Since HI 21cm absorption arises in cold neutral gas, this indicates that most gas-rich galaxies contain significant fractions of cold HI by $z \sim 1$. (abridged)
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