No matter which burst photo, until it isn't marked as 'Favorite' , it doesn't get to the actual 'Camera Roll'.
So, Mark those photos as 'Favorite'. Then, try connecting your iOS Device to your Mac. Now, the 'Image Capture' app should open automatically. [If it didn't, check the cable connection(USB<->iOS Device) or manually open the app. If you're not sure the problem is the wire(cable), see if the iOS device is being charged(of course this happens if you have turned on USB Charging in Settings).]
Also, there's a 3rd-party app named iExplorer which can help you in many ways. I recommend this app, but I'm not responsible for any further problems within your Macs or iOS devices.
As the iPhone User Guide (By Apple, available on the iBooks Store for free) suggests:
Take rapid-fire shots: (iPhone 5s) Touch and hold the shutter button to take photos in burst mode. The counter shows how many shots you’ve taken, until you lift your finger. To see the suggested shots and select the photos you want to keep, tap the thumbnail, then tap “Favorites.” The gray dot(s) mark the suggested photos. To copy a photo from the burst as a separate photo in Camera Roll, tap the circle on the lower-right corner. To delete the burst of photos, tap it, then tap .
