NDSolve with Piecewise solving BVP gives warning bvdisc

It's a backslide introduced since v9:

and another issue of "DiscontinuityProcessing" so we at least have 2 workarounds.

One is to turn off the "DiscontinuityProcessing":

eq1 = {y'[x] == Piecewise[{{z[x], x < 5}, {y[x], 8 > x > 5}, {2 z[x], x > 8}}]};
eq2 = {z'[x] == Piecewise[{{2 z[x] + 1, x < 5}, {2 y[x] + 1, x > 5}}]};
eqs = {eq1, eq2, y[0] == 1, z[10] == 0};
sol1 = NDSolve[eqs, {y, z}, {x, 0, 10}, Method -> {"DiscontinuityProcessing" -> False}]

But as pointed out by Michael E2 in his answer, this is dangerous.

Another workaround is to transform the Piecewise into UnitStep, with the help of the undocumented Simplify`PWToUnitStep:

eq1 = {y'[x] == Piecewise[{{z[x], x < 5}, {y[x], 8 > x > 5}, {2 z[x], x > 8}}]};
eq2 = {z'[x] == Piecewise[{{2 z[x] + 1, x < 5}, {2 y[x] + 1, x > 5}}]};
eqs = {eq1, eq2, y[0] == 1, z[10] == 0};
sol1 = NDSolve[eqs // Simplify`PWToUnitStep, {y, z}, {x, 0, 10}]

Interesting: Changing the conditions in the Piecewise function to simple one-sided inequalities allows NDSolve to work.

eq1 = {y'[x] == Piecewise[{{z[x], x < 5}, {2 z[x], x > 8}, {y[x], x > 5}}]};
eq2 = {z'[x] == Piecewise[{{2 z[x] + 1, x < 5}, {2 y[x] + 1, x > 5}}]};
eqs = {eq1, eq2, y[0] == 1, z[10] == 0};
sol1 = NDSolve[eqs, {y, z}, {x, 0, 10}, 
  Method -> {"Shooting", 
    "StartingInitialConditions" -> {y[0] == 1, z[0] == 1}}]

You still get a NDSolve::berr message (scaled boundary error), but that it not surprising in a discontinuous BVP.