Intersection of two surfaces

Show[ContourPlot3D[{z==y^2-x^2,x^2+y^2==1},{x,-2,2},{y,-2,2},{z,-2,2}],ParametricPlot3D[{Cos[t],Sin[t],Sin[t]^2-Cos[t]^2},{t,0,2Pi},PlotStyle->{Black,Thickness[.02]}]]

Surface bounded by the intersection

Splot=Show[Plot3D[y^2-x^2,{x,-1,1},{y,-Sqrt[1-x^2],Sqrt[1-x^2]},AxesLabel->{x,y,z},AspectRatio->1],ParametricPlot3D[{Cos[t],Sin[t],Sin[t]^2-Cos[t]^2},{t,0,2Pi},PlotStyle->{Black,Thickness[.02]}]]

Showing the normal vectors for orientation

Show[{Splot,Graphics3D[Table[{Arrowheads[Medium],Arrow[{{x,y,y^2-x^2},{x,y,y^2-x^2}+{2x,-2y,1}/Sqrt[4x^2+4y^2+1]/3}]},{x,-1,1,.2},{y,-Sqrt[1-x^2],Sqrt[1-x^2],.2}]]},PlotRange->All]