java.time
The java.util Date-Time API and their formatting API, SimpleDateFormat are outdated and error-prone. It is recommended to stop using them completely and switch to the modern Date-Time API*.
Also, quoted below is a notice from the home page of Joda-Time:
Note that from Java SE 8 onwards, users are asked to migrate to java.time (JSR-310) - a core part of the JDK which replaces this project.
Solution using java.time, the modern Date-Time API:
You do not need to form the string: You can use LocalDateTime#of to create an instance of LocalDateTime which can be converted into an Instant in order to get the number of milliseconds from the Unix epoch.
import java.time.Instant;
import java.time.LocalDateTime;
import java.time.ZoneOffset;
public class Main {
    public static void main(String[] args) {
        int year = 1970, month = 1, dayOfMonth = 1, hour = 8, minute = 0;
        LocalDateTime ldt = LocalDateTime.of(year, month, dayOfMonth, hour, minute);
        Instant instant = ldt.toInstant(ZoneOffset.UTC);
        System.out.println(instant.toEpochMilli());
    }
}
Output:
28800000
ONLINE DEMO
If you already have a date-time string in the given format:
import java.time.Instant;
import java.time.LocalDateTime;
import java.time.ZoneOffset;
import java.time.format.DateTimeFormatter;
import java.util.Locale;
public class Main {
    public static void main(String[] args) {
        DateTimeFormatter dtf = DateTimeFormatter.ofPattern("M/d/u h:m a", Locale.ENGLISH);
        String strDateTime = "1/1/1970 8:00 AM";
        LocalDateTime ldt = LocalDateTime.parse(strDateTime, dtf);
        Instant instant = ldt.toInstant(ZoneOffset.UTC);
        System.out.println(instant.toEpochMilli());
    }
}
Output:
28800000
ONLINE DEMO
An Instant represents an instantaneous point on the timeline in UTC.
Learn more about the modern Date-Time API from Trail: Date Time.
* For any reason, if you have to stick to Java 6 or Java 7, you can use ThreeTen-Backport which backports most of the java.time functionality to Java 6 & 7. If you are working for an Android project and your Android API level is still not compliant with Java-8, check Java 8+ APIs available through desugaring and How to use ThreeTenABP in Android Project.