We present new Goddard High-Resolution Spectrograph (GHRS) observations of the Ly alpha and Mg II absorption lines seen toward the nearby stars 61 Cyg A and 40 Eri A. We use these data to measure interstellar properties along these lines of sight and to search for evidence of circumstellar hydrogen walls, which are produced by collisions between the stellar winds and the local interstellar medium (LISM). We were able to model the Ly alpha lines of both stars without hydrogen-wall absorption components, but for 61 Cyg A the fit required a stellar Ly alpha line profile with an improbably deep self-reversal, and for 40 Eri A the fit required a very low deuterium-to-hydrogen ratio that is inconsistent with previous GHRS measurements. Since these problems could be rectified simply by including stellar hydrogen-wall components with reasonable attributes, our preferred fits to the data include these components. We have explored several ways in which the hydrogen-wall properties measured here and in previous work can be used to study stellar winds and the LISM. We argue that the existence of a hydrogen wail around 40 Eri A and a low II I column density along that line of sight imply that either the interstellar density must decrease toward 40 Eri A or the hydrogen ionization fraction (x) must increase. We find that hydrogen-wall temperatures are larger for stars with faster velocities through the LISM. The observed temperature-velocity relation is consistent with the predictions of hydromagnetic shock jump conditions. More precise comparison of the data and the jump conditions suggests crude upper limits for both x and the ratio of magnetic to thermal pressure in the LISM (alpha): x < 0.6 and alpha < 2. The latter upper limit corresponds to a limit on the LISM magnetic field of B < 5 mu G. These results imply that the plasma Mach number of the interstellar wind flowing into the heliosphere is M(A) > 1.3, which indicates that the collision is supersonic and that there should therefore be a bow shock outside the heliopause in the upwind direction. Finally, we estimate stellar wind pressures (P(wind)) from the measured hydrogen-wall column densities. These estimates represent the first empirical measurements of wind properties for late-type main-sequence stars. The wind pressures appear to be correlated with stellar X-ray surface fluxes, F(X), in a manner consistent with the relation P(wind) proportional to F(X)(-1/2), a relation that is also consistent with the variations of P(wind) and F(X) observed during the solar activity cycle. If this relation can in fact be generalized to solar-like stars, as is suggested by our data, then it is possible to estimate stellar wind properties simply by measuring stellar X-rays. One implication of this is that stellar wind pressures and mass-loss rates are then predicted to increase with time, since F(X) is known to decrease with stellar age.