My Freckle Face mini did the same thing two months ago, it dropped all of it's leaves after spider mites attacked it. I cut off all buds and left the canes about 6" tall, it has leaves and blooms now.

Give them water and sunlight and they should recover.

Take them outside. Roses are not house plants and they'll do much better outdoors. And one of the major problems with them inside is spider mies. If you have a patio or deck that gets good sun put them there. Or you can plant them in the ground.

Yes, the brown cane is probably dead. You can cut it off or possibly just pull it out. As long as the other canes stay green and moist they should leaf back out. Don't cut them back unless they turn black or brown. You need those leaf nodes along the green cane for it to leaf out. If you cut it to the soil it would have to send up all new canes and it may not have that much root ball to be able to do that yet. It might kill the plant entirely.

Keep them well watered but not soggy. If you have trays under the pots take them off. The pots have to be able to drain freely. The trays hold water up in the roots and they could rot. Do not fertilize them until they have new leaves. They don't need it yet and it could burn the roots. Once they leaf back out you can fertilize them at half strength. After they bloom you can go to full strength maybe once a month or every 6 weeks.

Thanks for your responses. I'm glad to hear that they'll recover. I'll place them outside on the deck and hopefully, that will help them recover fully.