The intersections are given by
Solve[u'[1 + x] == p, x]
Solve[p - p x/(2 epsH) == u'[1 + x], x, Reals]
You can plot them this way
Plot[{Derivative[1][u][1 + b[t]], p, p - (p b[t])/(2 epsH)}, {b[t], 0,
2 epsH},
Epilog -> {PointSize[Large],
Point[{x, u'[1 + x]} /.
Join[Solve[u'[1 + x] == p, x],
Solve[p - p x/(2 epsH) == u'[1 + x], x, Reals]]]}]
