Ethers has O in it so I expect them to have some polarity.
But, they doesn't have H atom (with partial positive charge) attached to O, they don't H-bond with themselves. However, their oxygen atom can H-bond with Hydrogen atom attached to any electronegative atom (mainly O,F,N)
So, yes Ether can form H bond with water and alcohol.
So hydrogen bonds love to form between acidic hydrogens and basic electron pairs. However, the only reason why ether may be able to make a hydrogen bond is if it represents the "basic electron pair" side. It would be unable to be the side which has the acidic hydrogens. Thus, I believe that if a hydrogen bond is formed, it will be that strong.
Side note: Most acidic hydrogens are usually on hydroxyls O-H