Do amines or alcohols have stronger intermolecular hydrogen bonds?
The boiling point of non-ionic compounds are highly depend on their H-bonding abilities. For example, boiling point of water (molar mass: $\pu{18.02 g/mol}$) is $\pu{100 ^\circ C}$ at $\pu{1 atm}$ while that of ethanol (molar mass: $\pu{46.07 g/mol}$) is $\pu{78.4 ^\circ C}$ at $\pu{1 atm}$, even though ethanol is heavier and have more other intermolecular attractions, excluding H-bonding, such as van der Waals interactions than water. This difference in boiling points displays clear dominamce of H-bonding (HB) on boiling point over other forces on these non-ionic compounds.

Therefore, since molar masses of methanol and methylamine is very close ($\pu{32.04 g/mol}$ for $\ce{CH3OH}$ versus $\pu{31.05 g/mol}$ for $\ce{CH3NH2}$), comparition of their boiling point would be excellent indicator of the comparison of their H-bonding strength: Boiling point of $\ce{CH3OH}$ is $\pu{64.7 ^\circ C}$ at $\pu{1 atm}$ while that of $\ce{CH3NH2}$  is $\pu{-6 ^\circ C}$ at $\pu{1 atm}$. Therefore, it is safe one to say that $\ce{CH3OH}$  has stronger H-bonding capabilities than that of $\ce{CH3NH2}$ even though $\ce{CH3NH2}$ has extra $\ce{H}$ compared to only one in $\ce{CH3OH}$. The best comparison measures of these two types of hydrogen bonding (HB) are the $\Delta H_\mathrm{HB}$ values in Wikipedia (Hydrogen Bonding): 


  
  $\Delta H_\mathrm{HB} = \pu{21 kJ/mol}$, listed for $\ce{O−H···:O}$ in water-water.
  $\Delta H_\mathrm{HB} = \pu{13 kJ/mol}$, listed for $\ce{N−H···:N}$ in ammonia-ammonia.