React Native Border Radius with background color

Try moving the button styling to the TouchableHighlight itself:

Styles:

submit: {
  marginRight: 40,
  marginLeft: 40,
  marginTop: 10,
  paddingTop: 20,
  paddingBottom: 20,
  backgroundColor: '#68a0cf',
  borderRadius: 10,
  borderWidth: 1,
  borderColor: '#fff',
},
submitText: {
  color: '#fff',
  textAlign: 'center',
}

Button (same):

<TouchableHighlight
  style={styles.submit}
  onPress={() => this.submitSuggestion(this.props)}
  underlayColor='#fff'>
    <Text style={[this.getFontSize(),styles.submitText]}>Submit</Text>
</TouchableHighlight>

You should add overflow: hidden to your styles:

Js:

<Button style={styles.submit}>Submit</Button>

Styles:

submit {
   backgroundColor: '#68a0cf';
   overflow: 'hidden';
}