The implicit differentiation gave you the function
(-7 - x - 7 x^2 - x^3)/(-24 - 2 y + y^2)
as the result for dy/dx. Notice that the numerator only depends on x and the denominator only depends on y. The horizontal tangents are where the expression is zero and this in turn is where the numerator is zero. So solve for the values of x where this is true:
Solve[-7 - x - 7 x^2 - x^3 == 0, x]
Once you have the x values from this (there is only one that is real in this case) you can find the corresponding y value by going back to the original implicit equation
84 x + 6 x^2 + 28 x^3 + 3 x^4 - 288 y - 12 y^2 + 4 y^3 == 5
substitute each x value and then solve for the corresponding y value for that x value. So now you have the {x,y} points where there are horizontal tangents.
Similarly the vertical tangents are where the expression goes to + or - infinity and this is where the denominator becomes zero (so long as it also is not one of the points where the numerator is also zero). You can determine this from
Solve[-24 - 2 y + y^2 == 0, y]
And like in the horizontal case, use the implicit equation to now solve for the x values corresponding to each of these y-values.
When you are done you can plot all of these values using ListPlot and then combine them with your original contour plot using Show.
You can make the points in your ListPlot show nicely by using something like (I am using made up points here):
ListPlot[{{1, 2}, {3, 5}},
PlotStyle -> Directive[Red, AbsolutePointSize[12]]]
If this ListPlot is called plot4 then you would combine this with your contour plot (which was called plot3) using
Show[plot3,plot4]
P.S. Give Professor Read our best regards ;-)