NSolve[{x, y, z} [Element]
   line && (surfaceequations1 || surfaceequations2 ||
    surfaceequations3) && ! (volumeequation1 &&
     volumeequation2) && ! (volumeequation2 &&
     volumeequation3) && ! (volumeequation1 && volumeequation3), {x,
  y, z}]

constraints[shapes__] :=
 And[## & @@ (Not /@
      Through[(RegionMember[RegionIntersection@##] & @@@
          Subsets[{shapes}, {2}])@#]),
   RegionMember[RegionUnion @@ (RegionBoundary /@ {shapes})]@#] &


intersections[l_, s__] :=
 NSolve[{x, y, z} [Element] l && constraints[s][{x, y, z}], {x, y,
   z}]

shape1 = Cuboid[{5, 5, 1}, {6, 25, 3}];
shape2 = Cuboid[{6, 5, 1}, {21, 6, 3}];
shape3 = Cuboid[{6, 24, 1}, {21, 25, 3}];
shapes = {shape1, shape2, shape3};
line = InfiniteLine[{{5.5, 10.5, 0.5}, {5.5, 10.5, 1}}];
intersection = intersections[line, ##] & @@ shapes;
intersection = DeleteDuplicates[intersection];
points = {x, y, z} /. intersection;
Graphics3D[{line, shapes, {Red, PointSize[0.05], Point[points]}}]

shape1 = Cuboid[{5, 5, 1}, {6, 25, 3}];
shape2 = Cuboid[{6, 5, 1}, {21, 6, 3}];
shape3 = Cuboid[{6, 24, 1}, {21, 25, 3}];
shape4 = Cuboid[{6, 6, 1}, {13, 13, 3}];
shape5 = Cuboid[{6, 17, 1}, {13, 24, 3}];
shapes = {shape1, shape2, shape3, shape4, shape5};
line = InfiniteLine[{{5.5, 10.5, 0.5}, {5.5, 10.5, 1}}];
intersection = intersections[line, ##] & @@ shapes;
intersection = DeleteDuplicates[intersection];
points = {x, y, z} /. intersection;
Graphics3D[{line, shapes, {Red, PointSize[0.05], Point[points]}}]

shape1 = Cuboid[{5, 5, 1}, {6, 25, 3}];
shape2 = Cuboid[{6, 5, 1}, {21, 6, 3}];
shape3 = Cuboid[{6, 24, 1}, {21, 25, 3}];
shape4 = Cuboid[{6, 6, 1}, {13, 13, 3}];
shape5 = Cuboid[{6, 17, 1}, {13, 24, 3}];
shape6 = Cuboid[{14, 13, 1}, {15, 17, 3}];
shape7 = Ellipsoid[{21, 9, 2}, {5, 4, 1}];
shape8 = Cuboid[{16, 6, 1}, {21, 13, 3}];
shape9 = Ellipsoid[{21, 21, 2}, {5, 4, 1}];
shape10 = Cuboid[{16, 17, 1}, {21, 24, 3}];
shape11 = Cuboid[{21, 9, 1}, {26, 10, 3}];
shape12 = Cuboid[{21, 20, 1}, {26, 21, 3}];
shape13 = Cuboid[{21, 10, 1}, {26, 13, 3}];
shape14 = Cuboid[{21, 17, 1}, {26, 20, 3}];
shape15 = Cuboid[{26, 11, 1}, {27, 13, 3}];
shape16 = Cuboid[{27, 12, 1}, {28, 18, 3}];
shape17 = Cuboid[{26, 17, 1}, {27, 19, 3}];

line = InfiniteLine[{{5.5, 10.5, 0.5}, {5.5, 10.5, 1}}];

shapes = {shape1, shape2, shape3, shape4, shape5, shape6, shape7,
   shape8, shape9, shape10, shape11, shape12, shape13, shape14,
   shape15, shape16, shape17};

intersection = intersections[line, ##] & @@ shapes;
intersection = DeleteDuplicates[intersection]
points = {x, y, z} /. intersection;
Graphics3D[{line, shapes, {Red, PointSize[0.05], Point[points]}}]