Not a solution in R but this should do it:

samtools view your.bam | cut -f1 | sort | uniq > read_names

samtools view file.bam |cut -f 1 ?

I got this off of stack exchange; it's faster, because it doesn't sort

samtools view mine.bam | cut -f 1 | awk '!x[$0]++' > read.names.txt

If your bam file is very light, in R you can do :

library(Rsamtools)
bam <- scanBam("input.bam")
bam[[1]]$qname

Otherwise, command line answers are preferred