Bosnia keeper Begovic made superb reaction saves from Ahmed Elmohamady and Yannick Sagbo in the first half to keep his side in the game, again denying the former Sunderland man after the break.
Stoke ended the game on top, but found McGregor in equally impressive form as he twice denied Peter Crouch from point-blank range.
Either side could have moved into the top half with a win, but remain 12th and 13th respectively with a point apiece.
Steve Bruce's side were unchanged from Monday's draw at Swansea City, while Stoke handed starts to Stephen Ireland and Oussama Assaidi following goalscoring substitute appearances in their memorable win over Chelsea last weekend.
The hosts were in the ascendancy from the off but could not find a way past Begovic.
Elmohamady was the first to stretch the Bosnian with a dipping volley from 25 yards that the Stoke goalkeeper did well to tip over the crossbar.
The Egyptian was in fine form on the right as he surged past Marc Muniesa before cutting back to Sagbo, who touched the ball away from Ryan Shawcross and brought a smart stop from Begovic at his near post.
Curtis Davies saw a firm header fly wide and Tom Huddlestone drilled a long-ranger off-target with 10 minutes to go in the half as Hull turned the screw, but saw their pressure amount to nothing.
Stoke ended the first period with their best spell, but only had a ball flashed across the face of goal from Marko Arnautovic to show for it as the sides went in at the break goalless.
Jake Livermore pounced following some dithering in the Stoke defence early in the second half, but his left-footed effort bobbled wide and Shawcross flicked a right-wing corner wide in an open 10 minutes after the interval.
Elmohamady and Begovic renewed rivalries on the hour, the keeper making an excellent stop from the wing-back's firm header.
Livermore fizzed another effort from distance wide and the hosts had penalty appeals waved away as Sagbo's header flicked off Muniesa's elbow.
Ireland tucked home after Crouch had forced McGregor into a stunning save with 20 minutes to go, only to be denied by the offside flag.
McGregor again got down to his left quickly to keep Crouch out as Stoke entered the closing stages strongly, Shawcross nodding onto the bar from the resulting corner.
But neither side could find a killer blow as Stoke recorded a sixth consecutive away game in the league without a win, while Hull racked up a fifth clean sheet on home soil this season.
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