The answer is No. My source is Rod Gow's rather wonderful little paper "Real-valued characters and the Schur index". Let me quote from the introduction:
It may happen that all elements of a group are strongly real while the group possesses real-valued characters of Schur index 2. An example is provided by the central product of quaternion and dihedral groups of order 8.
(The connection between the Schur index and the equalling 2 is equivalent to Frobenius-Schur Indicator can also be found in the introduction to Gow's paper.)equalling -1.)
